WASHINGTON—Struggling German lender Deutsche Bank won’t drive the eurozone economy into the ground, a member of the European Central Bank’s governing council said Thursday, adding a fresh dose of calm into a case that has raised concerns about the continent’s ability to confront the struggles of Germany’s largest lender.

“I don’t think that problems related with one of the banks somehow can influence overall financial stability,” said Vitas Vasiliauskas, the head of Lithuania’s central bank, in an interview with The Wall...